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Prove: The law of demand holds if WA, Walras' law, homogeneity of degree 0, and homogeneity of degree 1 in wealth hold for Walrasian demand functions

We need to show that: $$ (p - p')\cdot(x(p,1) - x(p', 1)) \le 0. $$ Note that if $x(p,1) = x(p', 1)$ then this is obviously satisfied, so assume that $x(p,1) \ne x(p', 1)$. Note that the condition is ...
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Proof for Marshallian Demand function

Let $x(p,w)$ be the demand at prices $p$ and income $w$. Let $x_0 = x(p_0,w)$ and $x_1 = x(p_1, w)$. Note that $p_0 x_0 = w = p_1 x_1$ Assume that WARP is violated, so $x_1 \ne x_0$, $$ p_0 x_0 \ge ...
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